Earlier quoted context omitted.
(Shrug) Put it in Antarctica. It'll be fine for 10,000 years there. No one will stumble across it accidentally, and we can get it back if we need it. I've never understood why this is considered a problem.
What gives you the ability to make a prediction on the 10,000-year scale? That seems wildly inappropriate. 10,000 years ago, you'd be the person saying "We should bury our magic artifacts in a big tomb in the desert! Nobody will find out what is inside for 10,000 years, because nobody can live in the desert and the sand is too hard to travel across..." Any you'd be completely wrong about that. Because 10,000 years is…
And it's unreasonable to suggest that nuclear waste couldn't be safely stored on the moon. Any future civilizations who can travel to the moon will almost certainly be acquainted with nuclear physics themselves. However, launching high-level waste into space has obvious risks of its own. It will also make it hard to recycle it later, if future technologies are able to extract more power from it.